File - math with mrs. Alligood

PreCalculus Notes
I.
POLAR FORM OF CONICS
Definition of Conics:
The set of points in the plane that move so that its distance from a fixed point (focus) is in constant ratio to its
distance from a fixed line (directrix) is a conic. The constant ratio is the eccentricity of the conic and is denoted by
e.
The conic is an ellipse if e < 1, a parabola if e = 1, and a hyperbola if e > 1.
II.
r=
Polar Equations of Conics:
ep
1 + e cos Ѳ
or
r=
ep____
1 + e sin Ѳ
e > 0 is the eccentricity and |p| is the distance between the pole (origin) and the directrix.
Nice Things to Know:
If the denominator is 1 + e sin Ѳ, it has a horizontal directrix above the pole.
If the denominator is 1 - e sin Ѳ, it has a horizontal directrix below the pole.
If the denominator is 1 + e cos Ѳ, it has a vertical directrix to the right of the pole.
If the denominator is 1 - e cos Ѳ, it has a vertical directrix to the left of the pole.
III.
Examples:
Guided Example 1: Using the given equation, determine the type of conic and state the location of the directrix:
𝟔
𝒓 = 𝟏−𝟐𝒄𝒐𝒔𝜽




The first # in the denominator is a 1, so this problem is already in the correct format.
The number in front of cosine is a 2. This means e = 2. e > 1, so this is a HYPERBOLA.
The numerator represents “ep”, so since e = 2, p must = 3. This means the directrix is 3 units away from
the pole. We need to decide if it is left/right/above/below.
Since the denominator has “ – cosine”, we know the directrix is to the LEFT of the pole (a focal point).
ANSWER: This is a hyperbola with the directrix located 3 units to the left of the pole.
Guided Example 2: r =


3_____
4 + 2 sin Ѳ
Which conic section is it? Divide so the first number in the denominator is 1. Check e.
Divide everything in the fraction by 4 so the equation will be in the correct form to pick out the parts.
r=
¾_______
1 + (1/2) sin Ѳ

Because of the “ + sin Ѳ” in the denominator, the graph has a horizontal directrix above the pole.
e = 1/2 so the graph is an ellipse.
𝟏
ep = 3/4 means 𝟐 p = 3/4 so p = 3/2
ANSWER: The graph is an ellipse with directrix 3/2 units above the pole (a focal point).
Guided Example 3: Find the equation with e = 1; directrix x = -1

The graph has a vertical directrix, so the equation will use cos Ѳ.
|p| = 1 (the distance between the pole and directrix)
ANSWER:
r=
1___
1 - cos Ѳ
Guided Example 4: Find the equation with e = 3/4; directrix y = -2

The graph has a horizontal directrix, so the equation will use sin Ѳ.
|p| = 2 (the distance between the pole and directrix)
ANSWER:
r=
3/2
(multiply all by 4) =
1 - (3/4) sin Ѳ
6
4 - 3 sin Ѳ
You try:
For the following, state the value of e, identify the type of conic, and describe the location of the directrix.
1.
r
8
1  2 cos 
2.
r
8
2  2 sin 
3.
r
4
3  sin 
4.
r
5
4  6 cos 
Find the polar equation of the conic described:
5. Focus is at the pole, eccentricity is 2/5 and the equation of the directrix is x = - 3.
6. Eccentricity, e = 3 and the equation of directrix is y = - 2.